Discrete Optimization
Tabu search for single machine scheduling with distinct due windows and weighted earliness/tardiness penalties

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Abstract

A single machine scheduling problem with distinct due windows to minimize total weighted earliness and tardiness is examined. A mathematical formulation of the problem is first presented and several important properties are studied to facilitate the solution process. An optimal timing algorithm is then proposed to determine completion times for each job in a given job sequence. A Tabu search (TS) procedure is employed together with the optimal timing algorithm to generate job sequences and final schedules. Computational experiments indicate that the performance of the proposed approach is quite well, especially for the instances of large size.

Introduction

With growing research interest in just-in-time (JIT) manufacturing, practitioners and researchers have recognized that in a schedule, a job finished either tardily or early incurs costs. This led to the consideration of both job earliness and tardiness as penalties in the objective function of a schedule. A large body of literature on this topic has appeared in last decade. However, most of the research focuses only on problems with a common due date/window rather than distinct due dates/windows (see [2]).

The single machine weighted earliness/tardiness scheduling problem with distinct due dates and no inserted idle time was studied by Adbul-Razaq and Potts [1], Ow and Morton [15], Li [13], and Liaw [14], among others. Adbul-Razaq and Potts [1] developed a Branch and Bound algorithm whose lower bound is obtained by performing recursions on a relaxed state space. The relaxed state space is generated by mapping states representing job subsets onto states representing total job processing time in the subsets. Ow and Morton [15] developed several heuristics and a filtered beam search algorithm, which, however, cannot guarantee the optimality of the solutions. On the other hand, Li [13] and Liaw [14] proposed Lagrangian relaxation based Branch and Bound algorithms that guarantee the optimality of the solution but require a great deal of computational effort.

The general cases of this problem are those that allow inserted idle times together with distinct due dates. As Baker and Scudder [2] pointed out, the assumption of no inserted idle times is inconsistent with the earliness/tardiness criterion because earliness is an irregular performance measure. Because of the hardness of the problems with idle times, Branch and Bound algorithm and search-based procedures become two typical approaches for these problems. In the former case, Hoogeveen and van de Velde [10] considered a single machine scheduling problem to minimize total weighted earliness and flow time, and proposed several estimation methods of lower bound for their Branch and Bound algorithm. However, the algorithm can only be used for problems with less than 20 jobs. In the latter case, procedures for this generalized problem normally consist of two parts, namely sequencing and optimal timing. Optimal timing algorithms for a given job sequence were discussed in Fry et al. [5], Garey et al. [6], and Yano and Kim [22]. Fry et al. [5] formulated the problem in a special linear programming model and Yano and Kim [22] adopted a dynamic programming formulation, but both approaches require large computation. Davis and Kanet [4], Szwarc and Mukhopadhyay [20], and Lee and Choi [12] independently presented optimal timing algorithms for problems with general due penalty weights. These algorithms focus on the weight comparison and were used in Branch and Bound algorithms or local search algorithms.

Algorithms for optimal job sequencing are relatively less studied. Yano and Kim [22] studied several dominance properties for sequencing jobs with penalty weights proportional to processing times. Szwarc [19] proposed a Branch and Bound algorithm based on some adjacent ordering conditions for jobs with distinct penalty weights. Lee and Choi [12] adopted a genetic algorithm for the same type of problems.

In many practical situations, jobs have distinct due windows rather than due dates because of uncertainty and tolerance. Koulamas [11] considered the problem with equal earliness/tardiness penalty weights and distinct due windows. He defined the earliness as the difference between the beginning of the due window and the starting time of a job (his definition of earliness is different from normal definition of earliness), and developed an optimal timing algorithm together with several sequencing heuristics.

This paper deals with a single machine scheduling problems where jobs have distinct due windows and inserted idle times in a schedule are allowed. Since several special cases (such as the single machine total weighted tardiness scheduling problem [2]) of this problem are strongly NP-hard, obviously it is strongly NP-hard, too.

In what follows, we propose an approach to combine a Tabu search (TS) procedure and an optimal timing algorithm for solving this problem. The procedure can be briefly described as follows (see Fig. 1):

  • (1) Generate an initial sequence, then go to step 3;

  • (2) Generate a job sequence by TS;

  • (3) Generate a schedule by optimal timing algorithm. If the stopping criterion is satisfied, then stop; otherwise, go to step 2.


The reminder of the paper is organized as follows. In Section 2, a mathematical description is given and three important properties of the problem are examined. Section 3 is devoted to the development of an optimal timing algorithm. A TS procedure based on sequencing properties is presented in Section 4. Section 5 contains the computational results of the proposed TS procedure and the comparison of the results with those from an incumbent Branch and Bound algorithm. Section 6 is a summary of the paper.

Section snippets

Problem description and the properties

Consider a single machine scheduling problem with distinct due windows and arbitrary earliness and tardiness penalty weights, under the assumptions that no preemption of the job processing is allowed and all the jobs are available at time zero. For convenience, we list the notations used throughout the paper as follows (without loss of generality, assume that all the parameters are non-negative integers):

  • N={1,2,…,n}: the set of jobs to be processed on the machine;

  • Π: the set of all schedules,

The optimal timing algorithm

The solution procedure for this scheduling problem consists of two parts: optimal timing and optimal sequencing. In this section we develop an optimal timing algorithm for a given job sequence by locating each job j at a completion time tj⩾0.

For the cases of distinct due dates, Garey et al. [6] proposed an O(nlogn) optimal timing algorithm for problems with symmetric earliness/tardiness penalty weights. Fry et al. [5] used a linear programming procedure, whereas Yano and Kim [22] used a dynamic

Sequencing by TS

In this section, we describe the TS based sequencing procedure for solving the problem. TS, initially proposed by Glover [7], [8], is an iterative improvement procedure designed to avoid local optimum for hard combinatorial optimization problems. It has been refined in Glover and Laguna [9], and several successful applications can be found in Reeves [18] and Rayward-Smith et al. [17]. The ideas of TS can be briefly sketched as follows. Starting from an initial solution, TS iteratively moves

Computational experiments

The TS procedure was implemented in Borland C++ on a PC (Intel PentiumII 266). We tested the algorithm on randomly generated problem instances and compared the results with those obtained by a Branch and Bound algorithm (incumbent solutions).

Conclusions

In this paper, we studied a single machine scheduling problem with distinct due windows to minimize the total weighted earliness and tardiness. We presented a mathematical formulation and three important properties of this problem. We also extended the optimal timing algorithm for a given sequence of the scheduling problem with distinct due dates to the case with distinct due windows. Based on these studies, we proposed a sequencing algorithm based on TS together with the optimal timing

Acknowledgements

Part of the work was done while G. Wan was a Ph.D. student at Hong Kong University of Science and Technology. The authors also would like to thank two anonymous referees for their helpful comments.

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